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A367725 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5) such that A(0) = 1.

%I A367725 #11 Mar 28 2025 06:07:27
%S A367725 1,1,5,30,205,1525,12001,98229,827651,7130614,62528631,556247554,
%T A367725 5007588460,45535148222,417625550140,3858724742014,35884576665516,
%U A367725 335616614245440,3154800011439675,29789198944740050,282426795122071741,2687467779597815314,25658105671446219050
%N A367725 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5) such that A(0) = 1.
%F A367725 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A367725 (1) x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5).
%F A367725 (2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^5 such that A(0) = 1.
%F A367725 (3) A(x) = x / Series_Reversion(x*(1 + Series_Reversion( x/((1 + x)^5 + x) ))).
%F A367725 (4) a(n) = (Sum_{k=1..n} binomial(n, k) * binomial(5*k-n, k-1))/n for n > 0 with a(0) = 1 (derived from a formula by _Tani Akinari_ in A243156).
%e A367725 G.f. A(x) = 1 + x + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12001*x^6 + 98229*x^7 + 827651*x^8 + 7130614*x^9 + 62528631*x^10 + ...
%e A367725 Let R(x) = x * (1 - x) / (1 - x - x^5) then R(A(x)) = x;
%e A367725 however, A(R(x)) does not equal x, rather
%e A367725 A(R(x)) = 1 + x + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12002*x^6 + 98240*x^7 + 827752*x^8 + 7131535*x^9 + 62537177*x^10 + ...
%o A367725 (PARI) {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^5 + x +x*O(x^n))))), n)}
%o A367725 for(n=0, 30, print1(a(n), ", "))
%o A367725 (PARI) /* From a formula by Tani Akinari in A243156 */
%o A367725 {a(n) = 0^n + sum(k=1, n, binomial(n, k)*binomial(5*k-n, k-1))/(n+0^n)}
%o A367725 for(n=0,30,print1(a(n),", "))
%Y A367725 Cf. A243156, A367724.
%K A367725 nonn
%O A367725 0,3
%A A367725 _Paul D. Hanna_, Nov 28 2023